Internal problem ID [6202]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.5. Exact Equations. Page
20
Problem number: 21.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\[ \boxed {\frac {4 y^{2}-2 x^{2}}{4 x y^{2}-x^{3}}+\frac {\left (8 y^{2}-x^{2}\right ) y^{\prime }}{4 y^{3}-y x^{2}}=0} \]
✓ Solution by Maple
Time used: 0.438 (sec). Leaf size: 225
dsolve(( (4*y(x)^2-2*x^2)/(4*x*y(x)^2-x^3))+( (8*y(x)^2-x^2)/(4*y(x)^3-x^2*y(x)) )*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {-c_{1} x -\frac {-2 c_{1}^{2} x^{2}+\sqrt {2 x^{4} c_{1}^{4}-2 \sqrt {c_{1}^{6} x^{6}+16}\, c_{1} x}}{2 c_{1} x}}{2 c_{1}} y \left (x \right ) = \frac {-c_{1} x -\frac {-2 c_{1}^{2} x^{2}+\sqrt {2 x^{4} c_{1}^{4}+2 \sqrt {c_{1}^{6} x^{6}+16}\, c_{1} x}}{2 c_{1} x}}{2 c_{1}} y \left (x \right ) = \frac {-c_{1} x +\frac {2 c_{1}^{2} x^{2}+\sqrt {2 x^{4} c_{1}^{4}-2 \sqrt {c_{1}^{6} x^{6}+16}\, c_{1} x}}{2 c_{1} x}}{2 c_{1}} y \left (x \right ) = \frac {-c_{1} x +\frac {2 c_{1}^{2} x^{2}+\sqrt {2 x^{4} c_{1}^{4}+2 \sqrt {c_{1}^{6} x^{6}+16}\, c_{1} x}}{2 c_{1} x}}{2 c_{1}} \end{align*}
✓ Solution by Mathematica
Time used: 12.331 (sec). Leaf size: 297
DSolve[( (4*y[x]^2-2*x^2)/(4*x*y[x]^2-x^3))+( (8*y[x]^2-x^2)/(4*y[x]^3-x^2*y[x]) )*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {x^2-\frac {\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} y(x)\to \frac {\sqrt {x^2-\frac {\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} y(x)\to -\frac {\sqrt {\frac {x^3+\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} y(x)\to \frac {\sqrt {\frac {x^3+\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} y(x)\to -\frac {\sqrt {x^2-\frac {\sqrt {x^6}}{x}}}{2 \sqrt {2}} y(x)\to \frac {\sqrt {x^2-\frac {\sqrt {x^6}}{x}}}{2 \sqrt {2}} y(x)\to -\frac {\sqrt {\frac {\sqrt {x^6}+x^3}{x}}}{2 \sqrt {2}} y(x)\to \frac {\sqrt {\frac {\sqrt {x^6}+x^3}{x}}}{2 \sqrt {2}} \end{align*}